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Fibonacci-like polynomials, the roots of which are responsible for a cyclic behavior of orbits of a second-order two-parametric difference equation, are considered. Using <i>Maple</i> and <i>Wolfram Alpha</i>, the location of the largest and the smallest roots responsible for the cycles of period <i>p</i> among the roots responsible for the cycles of periods 2<i>^kp</i> (period-doubling) and <i>kp</i> (period-multiplying) has been determined. These purely computational results of experimental mathematics, made possible by the use of modern digital tools, can be used as a motivation for confirmation through not-yet-developed methods of formal mathematics.